i.e., the average probability that the imperfect clones \(P_\alpha \)  and \(P_\beta \)  pass the corresponding test measurements for the ideal clones, \(M_{aa}\)  and \(M_{bb}\)  respectively. In quantum theory, the optimal cloning unitary achieves [1]}
\(F^{\rm Q,opt}_{g} &:= \frac{1}{4} \left[ \sqrt{(1+c_{ab})(1+ \sqrt{c_{ab}})} \right. \nonumber \\&\qquad \qquad \left.+ \sqrt{(1-c_{ab})(1-\sqrt{c_{ab}})}\right]^2,\) 
